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If mathematics are historical artifacts created by us, then how come unrelated theories end up been interderivable when they speak about completely unrelated subjects? E.g Lowenheim-Skolem-Tarski Theorem, Tychonoff's Theorem and Zorn's Lemma are all interderivable. If all mathematical entities are historical artifacts or inventions, how do mathematicians ever agree? How come mathematicians with absolutely no relation with each other who describe completely different and unrelated things end up with statements that can be derived from one another? This points to a deeper structure of reality that can not be explained by the "creation of mathematicians". To say that Mathematics is an invention is to completely ignore its dynamics and history. Mathematics points to a deeper structure of reality and yes there are mathematical facts which can be proven and also (some of them) can be applied to structures of the physical world, but also to any structure, which makes mathematics universal for any structured set of phenomena. That is completely different than saying that everything is reducible to mathematics, it is tantamount to saying that whatever object or set of objects that has a structure can be described mathematically. I am sorry to say this, but the arguments of the philosophers of this debate are pretty lame.

Cliff Harveyon06/11/2013 12:28am

Sorry to have to go there, but does Peter Hacker really know anything at all about math? What kinds of math courses has he ever taken? High school calculus? Somehow I doubt he even got that far.

His position of mathematical denialism is patently absurd. The whole point and definition of math is to uncover universally valid facts about logical structures. Once you have proven that "A implies B" it is then forever an undeniable fact that any structures that satisfy or express property A in any possible universe, must also express property B. Sorry, but to think that these structures exist only due to human beings is to completely fail to understand what math is. Hasn't he seen the movie Contact, where we communicate with an alien civilization by our common recognition of a universally important structure, the sequence of prime numbers?

The only thing that you can even argue depends on humans at all is the choice of which directions to look in and what structures are important. Emphasis may be human-dependent but mathematical truth is demonstrably not.

If even mathematics "doesn't exist" then certainly the entire body of work of Peter Hacker would have no hope to be regarded as any universal truth, as opposed to merely a fleeting product of a human imagination. If the space of logically provable propositions is "unreal", then certainly some ill-defined subjective opinion is infinitely more unreal. I'm genuinely curious what Hacker thinks he's doing exactly. Does he also regard himself as a mere "artisan"? If so I could at least admire his consistency, but it doesn't seem likely because based on what he says, logic is not a constraint on truth but instead something more like a fashion trend.

As for the rest of the debate, it left me pretty unsatisfied due to the lack of anybody on the other side, like Max Tegmark, to advocate for a more fundamental role for mathematics in our conception of Nature. Now that would be a genuinely interesting topic to debate, but you can never get there if you can't even agree on basic properties of mathematics. Lee Smolin is also a poor representative of the the theoretical physics community, and without that perspective this debate couldn't have gotten very far.

Lucianon17/10/2012 7:44pm

Hi Johan

One idea we're considering for this year is 'Are There Laws of Nature?', which will tackle the issues Lee raises, so watch this space!

In the mean time you might be interested in other debates that explore the limits of scientific knowledge and the question of if and why we can understand reality at all: "Alchemy, Anarchy and Science", "The Ultimate Particle" and the "Ultimate Map of Reality". These can be found under Science and Technology and Metaphysics and Language respectively on the IaI TV homepage search box.

Johan_0_oon12/10/2012 12:21pm

Having at first been persuaded by the ideas of our creating the system of mathematics, I then found it odd that both the physicist and that Oxford philosopher spoke about nature being 'kind to us'. What does that mean? nature gently conforms to our systems through some mysterious divine providence?

Is there another debate anywhere that actually delves into this latter question? if not, do one about that - specifically. Is nature kind to us? for building humans who can create systems that fit withitn it rather than jarring with it?

WhereWithAllon04/10/2012 1:14pm

Peter Hacker's description of math as a human invention and not a human discovery surely applies not only to the status of mathematical entities but that of all scientific creativity. Are numbers any more of an invention than forces, particles, or any of the other concepts that underpin what we think of as reality? If gravity is just an invention I'd like to see someone make do without it...